次の3次方程式を解け。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3-4x^2}\def\teisuf{+}\def\teisu{8} \def\yakusu{\pm1,\ \pm2,\ \pm4,\ \pm8} \def\testsiki{% & \colMM{orange}{P(1) = 1-4+8=5}\\ & \colMM{orange}{P(-1) = -1-4+8=3}\\ & \colMM{orange}{P(2) = 8-16+8=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{2} \def\dainyusiki{% \begin{align*} P(\dainyu) &= 2^3-4 \cdot 2^2 +8\\ \colMM{green}{\Darr } &= 8-16+8\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x-2} \def\syou{x^2-2x-4} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\ \\ & P(x) = 0\ から\\ & \warusiki=0 または \syou=0\\ \\ & \begin{align*} \warusiki &= 0\\ x &= \dainyu \end{align*}\\ \\ & \syou = 0\\ & \begin{align*} x &= \dfrac{2 \pm \sqrt{4-4 \cdot (-4)}}{2}\\ &= \dfrac{2 \pm \sqrt{20}}{2}\\ &= \dfrac{2 \pm 2\sqrt{5}}{2}\\ &= \dfrac{2(1 \pm \sqrt{5})}{2} = 1 \pm \sqrt{5}\\ \end{align*}\\ \\ & したがって x = 2,\ 1\pm\sqrt{5} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{}\def\vb{-4} \def\vcf{+}\def\vc{0} \def\vdf{+}\def\vd{8} \def\wv{2} \def\ws{x-2} \def\kb{2} \def\tbf{}\def\tb{-2} \def\kc{-4} \def\tcf{}\def\tc{-4} \def\kd{-8} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3+4x^2+x}\def\teisuf{}\def\teisu{-6} \def\yakusu{\pm1,\ \pm2,\ \pm3,\ \pm6} \def\testsiki{% & \colMM{orange}{P(1) = 1+4+1-6=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{1} \def\dainyusiki{% \begin{align*} P(\dainyu) &= 1^3+4 \cdot 1^2 +1-6\\ \colMM{green}{\Darr } &= 1+4+1-6\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x-1} \def\syou{x^2+5x+6} \def\insubunkai{(x+2)(x+3)} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & \begin{align*} P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\ &= (\warusiki)\insubunkai\\ \end{align*}\\ \\ & P(x) = 0\ から\\ & \warusiki=0{\scriptsize または }x+2=0{\scriptsize または }x+3=0\\ \\ & したがって x = 1,\ -2,\ -3 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{+}\def\vb{4} \def\vcf{+}\def\vc{1} \def\vdf{}\def\vd{-6} \def\wv{1} \def\ws{x-1} \def\kb{1} \def\tbf{+}\def\tb{5} \def\kc{5} \def\tcf{+}\def\tc{6} \def\kd{6} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3+4x^2+5x}\def\teisuf{+}\def\teisu{2} \def\yakusu{\pm1,\ \pm2} \def\testsiki{% & \colMM{orange}{P(1) = 1+4+5+2=12}\\ & \colMM{orange}{P(-1) = -1+4-5+2=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{-1} \def\dainyusiki{% \begin{align*} P(\dainyu) &= (-1)^3+4 \cdot (-1)^2 +5 \cdot (-1)+2\\ \colMM{green}{\Darr } &= -1+4-5+2\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x+1} \def\syou{x^2+3x+2} \def\insubunkai{(x+1)(x+2)\\&=(x+1)^2(x+2)} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & \begin{align*} P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\ &= (\warusiki)\insubunkai\\ \end{align*}\\ \\ & P(x) = 0\ から\\ & \warusiki=0{\scriptsize または }x+2=0\\ \\ & したがって x = -1,\ -2\colMM{lightgray}{ (x=-1\ は重解)} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{+}\def\vb{4} \def\vcf{+}\def\vc{5} \def\vdf{+}\def\vd{2} \def\wv{-1} \def\ws{x+1} \def\kb{-1} \def\tbf{+}\def\tb{3} \def\kc{-3} \def\tcf{+}\def\tc{2} \def\kd{-2} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3-3x^2}\def\teisuf{+}\def\teisu{2} \def\yakusu{\pm1,\ \pm2} \def\testsiki{% & \colMM{orange}{P(1) = 1-3+2=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{1} \def\dainyusiki{% \begin{align*} P(\dainyu) &= 1^3-3 \cdot 1^2+2\\ \colMM{green}{\Darr } &= 1-3+2\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x-1} \def\syou{x^2-2x-2} \def\insubunkai{(x+1)^2} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\ \\ & P(x) = 0\ から\\ & \warusiki=0 または \syou=0\\ \\ & \begin{align*} \warusiki &= 0\\ x &= \dainyu \end{align*}\\ \\ & \syou = 0\\ & \begin{align*} x &= \dfrac{2 \pm \sqrt{4-4 \cdot (-2)}}{2}\\ &= \dfrac{2 \pm \sqrt{12}}{2}\\ &= \dfrac{2 \pm 2\sqrt{3}}{2}\\ &= \dfrac{2(1 \pm \sqrt{3})}{2} = 1 \pm \sqrt{3}\\ \end{align*}\\ \\ & したがって x = 1,\ 1\pm\sqrt{3} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{}\def\vb{-3} \def\vcf{+}\def\vc{0} \def\vdf{+}\def\vd{2} \def\wv{1} \def\ws{x-1} \def\kb{1} \def\tbf{}\def\tb{-2} \def\kc{-2} \def\tcf{}\def\tc{-2} \def\kd{-2} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{2x^3-3x^2}\def\teisuf{}\def\teisu{-4} \def\yakusu{\pm1,\ \pm2,\ \pm4} \def\testsiki{% & \colMM{orange}{P(1) = 2-3-4=-5}\\ & \colMM{orange}{P(-1) = -2-3-4=-9}\\ & \colMM{orange}{P(2) = 16-12-4=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{2} \def\dainyusiki{% \begin{align*} P(\dainyu) &= 2 \cdot 2^3-3 \cdot 2^2-4\\ \colMM{green}{\Darr } &= 16-12-4\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x-2} \def\syou{2x^2+x+2} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\ \\ & P(x) = 0\ から\\ & \warusiki=0 または \syou=0\\ \\ & \begin{align*} \warusiki &= 0\\ x &= \dainyu \end{align*}\\ \\ & \syou = 0\\ & \begin{align*} x &= \dfrac{-1 \pm \sqrt{1-4 \cdot 4}}{4}\\ &= \dfrac{-1 \pm \sqrt{-15}}{4}\\ &= \dfrac{-1 \pm \sqrt{15}\,i}{4}\\ \end{align*}\\ \\ & したがって x = -1,\ \dfrac{-1 \pm \sqrt{15}\,i}{4} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{2} \def\vbf{}\def\vb{-3} \def\vcf{+}\def\vc{0} \def\vdf{}\def\vd{-4} \def\wv{2} \def\ws{x-2} \def\kb{4} \def\tbf{+}\def\tb{1} \def\kc{2} \def\tcf{+}\def\tc{2} \def\kd{4} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
因数分解の問題をリサイクル!
次の3次方程式を解け。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3+2x^2-x}\def\teisuf{}\def\teisu{-2} \def\yakusu{\pm1,\ \pm2} \def\testsiki{% & \colMM{orange}{P(1) = 1+2-1-2=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{1} \def\dainyusiki{% \begin{align*} P(\dainyu) &= 1^3+2 \cdot 1^2 -1-2\\ \colMM{green}{\Darr } &= 1+2-1-2\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x-1} \def\syou{x^2+3x+2} \def\insubunkai{(x+1)(x+2)} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & \begin{align*} P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\ &= (\warusiki)\insubunkai\\ \end{align*}\\ \\ & P(x) = 0\ から\\ & \warusiki=0{\scriptsize または }x+1=0{\scriptsize または }x+2=0\\ \\ & したがって x = \pm1,\ -2 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{+}\def\vb{2} \def\vcf{}\def\vc{-1} \def\vdf{}\def\vd{-2} \def\wv{1} \def\ws{x-1} \def\kb{1} \def\tbf{+}\def\tb{3} \def\kc{3} \def\tcf{+}\def\tc{2} \def\kd{2} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3-7x}\def\teisuf{}\def\teisu{-6} \def\yakusu{\pm1,\ \pm2,\ \pm3,\ \pm6} \def\testsiki{% & \colMM{orange}{P(1) = 1-7-6=-12}\\ & \colMM{orange}{P(-1) = -1+7-6=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{-1} \def\dainyusiki{% \begin{align*} P(\dainyu) &= (-1)^3-7 \cdot (-1) -6\\ \colMM{green}{\Darr } &= -1+7-6\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x+1} \def\syou{x^2-x-6} \def\insubunkai{(x+2)(x-3)} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & \begin{align*} P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\ &= (\warusiki)\insubunkai\\ \end{align*}\\ \\ & P(x) = 0\ から\\ & \warusiki=0{\scriptsize または }x+2=0{\scriptsize または }x-3=0\\ \\ & したがって x = -1,\ -2,\ 3 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{+}\def\vb{0} \def\vcf{}\def\vc{-7} \def\vdf{}\def\vd{-6} \def\wv{-1} \def\ws{x+1} \def\kb{-1} \def\tbf{}\def\tb{-1} \def\kc{1} \def\tcf{}\def\tc{-6} \def\kd{6} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
応用問題に登場!
次の3次方程式を解け。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3+x}\def\teisuf{+}\def\teisu{10} \def\yakusu{\pm1,\ \pm2,\ \pm5,\ \pm10} \def\testsiki{% & \colMM{orange}{P(1) = 1+1+10=12}\\ & \colMM{orange}{P(-1) = -1-1+10=8}\\ & \colMM{orange}{P(2) = 8+2+10=20}\\ & \colMM{orange}{P(-2) = -8-2+10=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{-2} \def\dainyusiki{% \begin{align*} P(\dainyu) &= (-2)^3+(-2)+10\\ \colMM{green}{\Darr } &= -8-2+10\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x+2} \def\syou{x^2-2x+5} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\ \\ & P(x) = 0\ から\\ & \warusiki=0 または \syou=0\\ \\ & \begin{align*} \warusiki &= 0\\ x &= \dainyu \end{align*}\\ \\ & \syou = 0\\ & \begin{align*} x &= \dfrac{2 \pm \sqrt{4-4 \cdot 5}}{2}\\ &= \dfrac{2 \pm \sqrt{-16}}{2}\\ &= \dfrac{2 \pm \sqrt{16}\,i}{2}\\ &= \dfrac{2 \pm 4\,i}{2}\\ &= \dfrac{2(1 \pm 2\,i)}{2} = 1 \pm 2\,i\\ \end{align*}\\ \\ & したがって x = -2,\ 1\pm2\,i \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{+}\def\vb{0} \def\vcf{+}\def\vc{1} \def\vdf{+}\def\vd{10} \def\wv{-2} \def\ws{x+2} \def\kb{-2} \def\tbf{}\def\tb{-2} \def\kc{4} \def\tcf{+}\def\tc{5} \def\kd{-10} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3+x^2-4x}\def\teisuf{+}\def\teisu{6} \def\yakusu{\pm1,\ \pm2,\ \pm3,\ \pm6} \def\testsiki{% & \colMM{orange}{P(1) = 1+1-4+6=4}\\ & \colMM{orange}{P(-1) = -1+1+4+6=10}\\ & \colMM{orange}{P(2) = 8+4-8+6=10}\\ & \colMM{orange}{P(-2) = -8+4+8+6=10}\\ & \colMM{orange}{P(3) = 27+9-12+6=30}\\ & \colMM{orange}{P(-3) = -27+9+12+6=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{-3} \def\dainyusiki{% \begin{align*} P(\dainyu) &= (-3)^3+(-3)^2-4 \cdot (-3)+6\\ \colMM{green}{\Darr } &= -27+9+12+6\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x+3} \def\syou{x^2-2x+2} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\ \\ & P(x) = 0\ から\\ & \warusiki=0 または \syou=0\\ \\ & \begin{align*} \warusiki &= 0\\ x &= \dainyu \end{align*}\\ \\ & \syou = 0\\ & \begin{align*} x &= \dfrac{2 \pm \sqrt{4-4 \cdot 2}}{2}\\ &= \dfrac{2 \pm \sqrt{-4}}{2}\\ &= \dfrac{2 \pm \sqrt{4}\,i}{2}\\ &= \dfrac{2 \pm 2\,i}{2}\\ &= \dfrac{2(1 \pm i)}{2} = 1 \pm i\\ \end{align*}\\ \\ & したがって x = -3,\ 1\pm i \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{+}\def\vb{1} \def\vcf{}\def\vc{-4} \def\vdf{+}\def\vd{6} \def\wv{-3} \def\ws{x+3} \def\kb{-3} \def\tbf{}\def\tb{-2} \def\kc{6} \def\tcf{+}\def\tc{2} \def\kd{-6} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\siki{x^3-3x^2+x}\def\teisuf{+}\def\teisu{5} \def\yakusu{\pm1,\ \pm5} \def\testsiki{% & \colMM{orange}{P(1) = 1-3+1+5=4}\\ & \colMM{orange}{P(-1) = -1-3-1+5=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた!} } \def\dainyu{-1} \def\dainyusiki{% \begin{align*} P(\dainyu) &= (-1)^3-3(-1)^2+(-1)+5\\ \colMM{green}{\Darr } &= -1-3-1+5\\ \colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが0・・・割り切れる!} \end{align*} } \def\warusiki{x+1} \def\syou{x^2-4x+5} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colFB{orange}{下調べ}\\ & \ \colMM{orange}{P(k)=0\ になる\ k\ を探す!}\\ & \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入!}\\ \testsiki\\ \\ & P(x) = \siki\teisuf\teisu\ とすると\\ & \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\ & \dainyusiki\\ & \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\ & よって,P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\ & \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り0}\\ & P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\ \\ & P(x) = 0\ から\\ & \warusiki=0 または \syou=0\\ \\ & \begin{align*} \warusiki &= 0\\ x &= \dainyu \end{align*}\\ \\ & \syou = 0\\ & \begin{align*} x &= \dfrac{4 \pm \sqrt{16-4 \cdot 5}}{2}\\ &= \dfrac{4 \pm \sqrt{-4}}{2}\\ &= \dfrac{4 \pm \sqrt{4}\,i}{2}\\ &= \dfrac{4 \pm 2\,i}{2}\\ &= \dfrac{2(2 \pm i)}{2} = 2 \pm i\\ \end{align*}\\ \\ & したがって x = -1,\ 2\pm i \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
組立除法
\def\va{1} \def\vbf{}\def\vb{-3} \def\vcf{+}\def\vc{1} \def\vdf{+}\def\vd{5} \def\wv{-1} \def\ws{x+1} \def\kb{-1} \def\tbf{}\def\tb{-4} \def\kc{4} \def\tcf{+}\def\tc{5} \def\kd{-5} \def\td{0} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{matrix} \va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\ \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\ & & & & \colMM{red}{\Darr}\\ \va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\ & \kb & \kc & \kd & \\\hline \va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\ \colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\ \va x^2 & \tb x & \tc\\ \end{matrix} \\ \begin{align*}\\ \colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\ & \ \colFR{black}{\scriptsize\bf 余} \ \td \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan